Orientador: Hélio Vasconcelos Fagundes / Mestre

In this thesis, we consider the problem of solving large and sparse linear systems of saddle point type stemming from optimization problems. The focus of the thesis is on iterative methods, and new preconditioning srategies are proposed, along with novel spectral estimtates for the matrices involved.

This Ph.D thesis focuses on iterative regularization methods for regularizing linear and nonlinear ill-posed problems. Regarding linear problems, three new stopping rules for the Conjugate Gradient method applied to the normal equations are proposed and tested in many numerical simulations, including some tomographic images reconstruction problems. Regarding nonlinear problems, convergence and convergence rate results are provided for a Newton-type method with a modified version of Landweber iteration as an inner iteration in a Banach space setting.

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Resumo: Satyrinae é uma das onze subfamílias de Nymphalidae, com cerca de 2.500 espécies descritas, distribuídas em 255 gêneros. Apesar de sua ampla distribuição mundial, cerca de 50% dos gêneros ocorrem na região Neotropical. Ainda há muitas controvérsias em relação à classificação das categorias mais baixas de Satyrinae, sendo cada vez mais necessários estudos taxonômicos para entender as delimitações dos gêneros e verificar se suas espécies estão devidamente alocadas. A partir da revisão taxonômica, Pampasatyrus Hayward, 1953 (Lepidoptera: Nymphalidae, Satyrinae, Pronophilini) é revalidado e um novo arranjo taxonômico para o gênero é proposto. Duas espécies novas são descritas, Pampasatyrus sp. n. 1, com ocorrência em São Paulo e Santa Catarina, Brasil, e Pampasatyrus sp. n. 2 de Minas Gerais e Rio de Janeiro, Brasil. Uma subespécie nova para Pampasatyrus reticulata é descrita para São Paulo, Brasil. São designados o neótipo de Neomaenas reticulata Weymer, 1907 e lectótipos de Epinephele gyrtone Berg, 1877, Epinephele nilesi Weeks, 1902 e Satyrus quies Berg, 1877. Três espécies são transferidas de Pampasatyrus para Gênero A, gen. n. (Euptychiina): Gênero A imbrialis (Weeks, 1901) comb. n. da Bolívia (Cochabamba), Gênero A ocelloides (Schaus, 1902) comb. n., com ocorrência no Paraguai (Hernandarias e Caaguazú) e Brasil (regiões centro-oeste, sul e sudeste) e Gênero A periphas (Godart, [1824]) comb. n., distribuídas desde o sul do Brasil até a região nordeste da Argentina (Buenos Aires). São designados lectótipos de Epinephele imbrialis Weeks, 1901 e de Euptychia ocelloides Schaus, 1902.

Orientador: Hélio V. Fagundes / Mestre

Orientador: Marcelo Moreira Ganzarolli / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecanica / Made available in DSpace on 2018-07-27T22:00:16Z (GMT). No. of bitstreams: 1 Mascarenhas_CristianoHenriquedeOliveira_M.pdf: 5440141 bytes, checksum: 9a4b0e7d84776c45c08f2c83a76cc843 (MD5) Previous issue date: 2001 / Resumo: Há várias aplicações onde as técnicas de geração de malhas são empregadas: na aeronin9-mica (cálculo de coeficientes de arrasto), em trocadores de calor (cálculo do número de Nusselt), etc. Neste trabalho buscou-se mostrar a eficácia das técnicas de geração de malha na solução de problemas de engenharia, com geometrias irregulares, como em banco de tubos, por exemplo. Nestes casos, uma malha ajustada à fronteira do corpo favorece a obtenção de soluções numéricas mais confiáveis. Das metodologias implementadas, a geração elíptica (que tem como base um sistema de equações diferenciais parciais não-lineares), permite controlar certas carecterísticas desejadas, como ortogonalidade e espaçamento (nas fronteiras). Neste caso, as equações originais são resolvidas no plano lógico. A obtenção das carecterísticas especiais, depende da implementação de certas funções de controle. Na técnica multibloco, divide-se o domínio principal em duas sub-malhas independentes, a polar e a cartesiana. O acoplamento destas exige uma rotina para passar dados entre as malh8.s, na região de contato, via interpolação. Buscou-se, aqui, compará-las, com o objetivo de aferir a importância da malha gerada na precisão do campo analisado (um problema de condução do calor bidimensional é usado nesta aferição). O laplaciano é resolvido em coordenadas generalizadas e nas coordenadas polar e cartesiana, usando-se o método de volumes finitos nas discretizações. Os resultados obtidos foram plotados, analisados e comparados com uma solução exata, tendo sido observadas tendências semelhantes (física e geométrica), mas ganho muito superior para a técnica elíptica / Abstract: There are many applications where one of the techniques here used are applied with relevance, for instance, into aerodynamics in the calculus of drag coefficient, in the heat exchangers (calculus of Nusselt number). In this work it was sougth to show the efficacy of grid generation techniques on the solution of engeneering problems. Specifically at one geometry type tube banks, by sample. In these cases a grid boundary fitted increases greatly the obtention of more reliable numerical solutions. In this context, two strategies for grid generation were developed. The elliptical approach (which has as basis a system of non-linear partial differential equations) implemented, permits to obtain automatically specific characteristics such as specified space and ortoghonality (both in the boundary). The equations are solved into transformed plane using TDMA solver. To obtain those conditions control functions are implemented. In the second strategy, multibloc technique, the main domain is divided into two independent sub-grids, one polar and other cartesian. The join of sub-grids demands a procedure for information transfer, on the contact boundary, by interpolation. In this study there was a focus to analyse the two strategies comparatively with the aim of gauging the importance of grid genaration on the accurancy over the field of temperature. A bidimensional heat transfer problem is used to check it out. The laplacian operator is solved in generalized, polar and cartesian coordinates, using finite volumes method into discretizations. The results obtained were plotted, analysed and compared with an exact solution, have been observed similar tendencies (physics and geometrics) but with stronger gains for the elliptic formulation / Mestrado / Termica e Fluidos / Mestre em Engenharia Mecânica

In this thesis we present a new and accurate series of computation methods for compressible multi-phase flows with capillary effects based upon the full seven-equation Baer-Nunziato model. For that reason, there are some numerical methods to obtain high accuracy solutions, which will be shown here. First, a high resolution shock capturing Total Variation Diminishing (TVD) finite volume scheme is used on both Cartesian and unstructured triangular grids. Regarding the TVD finite volume scheme on the unstructured grid, time-accurate local time stepping (LTS) is applied to compute the solutions of the governing PDE system, in which the results are also compared with time-accurate global time stepping. Second, we propose a novel high order accurate numerical method for the solution of the seven equation Baer-Nunziato model based on ADER discontinuous Galerkin (DG) finite element schemes combined with a posteriori subcell finite volume limiting and adaptive mesh refinement (AMR). In multi-phase flows, the difficulty is to design accurate numerical methods for resolving the phase interface, as well as the simulation of the phenomena occurring at the interface, such as surface tension effects, heat transfer and friction. This is because of the interactions of the fluids at the phase interface and its complex geometry. So the accurate simulation of compressible multi-phase flows with surface tension effects is currently still one of the most challenging problems in computational fluid dynamics (CFD). In this work, we present a novel path-conservative finite volume discretization of the continuum surface force method (CSF) of Brackbill et al. to account for the surface tension effect due to curvature of the phase interface. This is achieved in the context of a diffuse interface approach, based on the seven equation Baer-Nunziato model of compressible multi-phase flows. Such diffuse interface methods for compressible multi-phase flows including capillary effects have first been proposed by Perigaud and Saurel. Regarding the high order accuracy of a diffuse interface approach, the interface is captured and allowed to travel across one single possibly refined cell, and is computed in the context of multi-dimensional high accurate space/time DG schemes with AMR and a posteriori sub-cell stabilization. The surface tension terms of the CSF approach are considered as a part of the non-conservative hyperbolic system. We propose to integrate the CSF source term as a non-conservative product and not simply as a source term, following the ideas on path conservative finite volume schemes put forward by Castro and Parés. For the validation of the current numerical methods, we compare our numerical results with those published previously in the literature.

In this work the numerical discretization of the partial differential governing equations for compressible and incompressible flows is dealt within the discontinuous Galerkin (DG) framework along space-time adaptive meshes. Two main research fields can be distinguished: (1) fully explicit DG methods on collocated grids and (2) semi-implicit DG methods on edge-based staggered grids. DG methods became increasingly popular in the last twenty years mainly because of three intriguing properties: i) non-linear L2 stability has been proven; ii) arbitrary high order of accuracy can be achieved by simply increasing the polynomial order of the chosen basis functions, used for approximating the state-variables; iii) high scalability properties make DG methods suitable for large-scale simulations on general unstructured meshes. It is a well known fact that a major weakness of high order DG methods lies in the difficulty of limiting discontinuous solutions, which generate spurious oscillations, namely the so-called ’Gibbs phenomenon’. Over the years, several attempts have been made to cope with this problem and different kinds of limiters have been proposed. Among them, a rather intriguing paradigm has been defined in the work of [71], in which the nonlinear stabilization of the scheme is sequentially and locally introduced only for troubled cells on the basis of a multidimensional optimal order detection (MOOD) criterion. In the present work the main benefits of the MOOD paradigm, i.e. the computational robustness even in the presence of strong shocks, are preserved and the numerical diffusion is considerably reduced also for the limited cells by resorting to a proper sub-grid. In practice the method first produces a so-called candidate solution by using a high order accurate unlimited DG scheme. Then, a set of numerical and physical detection criteria is applied to the candidate solution, namely: positivity of pressure and density, absence of floating point errors and satisfaction of a discrete maximum principle in the sense of polynomials. Then, in those cells where at least one of these criteria is violated the computed candidate solution is detected as troubled and is locally rejected. Next, the numerical solution of the previous time step is scattered onto cell averages on a suitable sub-grid in order to preserve the natural sub-cell resolution of the DG scheme. Then, a more reliable numerical solution is recomputed a posteriori by employing a more robust but still very accurate ADER-WENO finite volume scheme on the sub-grid averages within that troubled cell. Finally, a high order DG polynomial is reconstructed back from the evolved sub-cell averages. Moreover, handling typical multiscale problems, dynamic adaptive mesh refinement (AMR) and adaptive polynomial order methods are probably the two main ways of preserving accuracy and efficiency, and saving computational effort. The here adopted AMRapproach is the so called ’cell by cell’ refinement because of its formally very simple tree-type data structure. In the here-presented ’cell-by-cell’ AMR every single element is recursively refined, from a coarsest refinement level l0 = 0 to a prescribed finest (maximum) refinement level lmax, accordingly to a refinement-estimator function X that drives step by step the choice for recoarsening or refinement. The combination of the sub-cell resolution with the advantages of AMR allows for an unprecedented ability in resolving even the finest details in the dynamics of the fluid. First, the Euler equations of compressible gas dynamics and the magnetohydrodynamics (MHD) equations have been treated [281]. Then, the presented method has been readily extended to the special relativistic ideal MHD equations [280], but also the the case of diffusive fluids, i.e. fluid flows in the presence of viscosity, thermal conductivity and magnetic resistivity [116]. In particular, the adopted formalism is quite general, leading to a novel family of adaptive ADER-DG schemes suitable for hyperbolic systems of partial differential equations in which the numerical fluxes also depend on the gradient of the state vector because of the parabolic nature of diffusive terms. The presented results show clearly that the shock-capturing capability of the news schemes are significantly enhanced within the cell-by-cell Adaptive Mesh Refinement (AMR) implementation together with time accurate local time stepping (LTS). The resolution properties of the new scheme have been shown through a wide number of test cases performed in two and in three space dimensions, from low to high Mach numbers, from low to high Reynolds regimes. In particular, concerning MHD equations, the divergence-free character of the magnetic field is taken into account through the so-called hyperbolic ’divergence-cleaning’ approach which allows to artificially transport and spread the numerical spurious ’magnetic monopoles’ out of the computational domain. A special treatment has been followed for the incompressible Navier-Stokes equations. In fact, the elliptic character of the incompressible Navier-Stokes equations introduces an important difficulty in their numerical solution: whenever the smallest physical or numerical perturbation arises in the fluid flow then it will instantaneously affect the entire computational domain. Thus, a semi-implicit approach has been used. The main advantage of making use of a semi-implicit discretization is that the numerical stability can be obtained for large time-steps without leading to an excessive computational demand [117]. In this context, we derived two new families of spectral semi-implicit and spectral space-time DG methods for the solution of the two and three dimensional Navier-Stokes equations on edge-based staggered Cartesian grids [115], following the ideas outlined in [97] for the shallow water equations. The discrete solutions of pressure and velocity are expressed in the form of piecewise polynomials along different meshes. While the pressure is defined on the control volumes of the main grid, the velocity components are defined on edge-based dual control volumes, leading to a spatially staggered mesh. In the first family, high order of accuracy is achieved only in space, while a simple semi-implicit time discretization is derived by introducing an implicitness factor theta in [0.5, 1] for the pressure gradient in the momentum equation. The real advantages of the staggering arise after substituting the discrete momentum equation into the weak form of the continuity equation. In fact, the resulting linear system for the pressure is symmetric and positive definite and either block penta-diagonal (in 2D) or block hepta-diagonal (in 3D). As a consequence, the pressure system can be solved very efficiently by means of a classical matrix-free conjugate gradient method. Moreover, a rigorous theoretical analysis of the condition number of the resulting linear systems and the design of specific preconditioners, using the theory of matrix-valued symbols and Generalized Locally Toeplitz (GLT) algebra has been successfully carried out with promising results in terms of numerical efficiency [102]. The resulting algorithm is stable, computationally very efficient, and at the same time arbitrary high order accurate in both space and time. The new numerical method has been thoroughly validated for approximation polynomials of degree up to N = 11, using a large set of non-trivial test problems in two and three space dimensions, for which either analytical, numerical or experimental reference solutions exist. Moreover, the here mentioned semi-implicit DG method has been successfully extended to a novel edge-based staggered ’cell-by-cell’ adaptive meshes [114].